Future Value
------------

The `b F' (`calc-fin-fv') [`fv'] command computes the future value of
an investment.  It takes three arguments from the stack: `fv(RATE, N,
PAYMENT)'.  If you give payments of PAYMENT every year for N years,
and the money you have paid earns interest at RATE per year, then this
function tells you what your investment would be worth at the end of
the period.  (The actual interval doesn't have to be years, as long as
N and RATE are expressed in terms of the same intervals.)  This
function assumes payments occur at the *end* of each interval.

The `I b F' [`fvb'] command does the same computation, but assuming
your payments are at the beginning of each interval.  Suppose you plan
to deposit $1000 per year in a savings account earning 5.4% interest,
starting right now.  How much will be in the account after five years?
`fvb(5.4%, 5, 1000) = 5870.73'.  Thus you will have earned $870 worth
of interest over the years.  Using the stack, this calculation would
have been `5.4 M-% 5 RET 1000 I b F'.  Note that the rate is expressed
as a number between 0 and 1, *not* as a percentage.

The `H b F' [`fvl'] command computes the future value of an initial
lump sum investment.  Suppose you could deposit those five thousand
dollars in the bank right now; how much would they be worth in five
years?  `fvl(5.4%, 5, 5000) = 6503.89'.

The algebraic functions `fv' and `fvb' accept an optional
fourth argument, which is used as an initial lump sum in the sense
of `fvl'.  In other words, `fv(RATE, N,
PAYMENT, INITIAL) = fv(RATE, N, PAYMENT)
+ fvl(RATE, N, INITIAL)'.

To illustrate the relationships between these functions, we could do
the `fvb' calculation "by hand" using `fvl'.  The final balance will
be the sum of the contributions of our five deposits at various times.
The first deposit earns interest for five years: `fvl(5.4%, 5, 1000) =
1300.78'.  The second deposit only earns interest for four years:
`fvl(5.4%, 4, 1000) = 1234.13'.  And so on down to the last deposit,
which earns one year's interest: `fvl(5.4%, 1, 1000) = 1054.00'.  The
sum of these five values is, sure enough, $5870.73, just as was
computed by `fvb' directly.

What does `fv(5.4%, 5, 1000) = 5569.96' mean?  The payments are now at
the ends of the periods.  The end of one year is the same as the
beginning of the next, so what this really means is that we've lost
the payment at year zero (which contributed $1300.78), but we're now
counting the payment at year five (which, since it didn't have a
chance to earn interest, counts as $1000).  Indeed, `5569.96 = 5870.73
- 1300.78 + 1000' (give or take a bit of roundoff error).